# How to Graph Solutions to One-step and Two-step Linear Inequalities

Dive into the mysterious world of linear inequalities, where the solutions often lie in a vast stretch of the number line rather than a single, fixed point. This journey will explore one-step and two-step inequalities, inviting you to master the nuances of their graphical representations.

## Step-by-Step Guide to Graphing Solutions to One-step and Two-step Linear Inequalities

Here is a step-by-step guide to graphing solutions to one-step and two-step linear inequalities:

### Step 1: Understand the Terrain: Basic Inequalities Overview

**Less than (\(<\))**: Think of a hungry alligator, always eager to chomp the smaller number.**Greater than (\(>\))**: The reverse. The alligator now wants the bigger number.**Less than or equal to (\(≤\))**: Here, the alligator doesn’t mind if it’s exactly equal or just a tad smaller.**Greater than or equal to (\(≥\))**: Big or just the same, both are good enough!

### Step 2: Setting the Stage: Drawing a Number Line

- Take a ruler or a straightedge.
- Sketch a horizontal line, which will represent our number line.
- Evenly space and mark numbers on this line. For instance, from \(-10\) to \(10\).

### Step 3: One-Step Inequalities: Baby Steps to Mastery

a) **Isolate the Variable**

i) If the inequality is \(x>5\), then \(x\) is already isolated.

ii) For an inequality like \(x+4<7\), subtract \(4\) from each side to get\(x<3\).

b) **Graph the Solution**

i) For **strict inequalities** like \(x<3\) or \(x>5\):

- Find the number on the number line.
- Make an open circle on it, indicating the value is not included.
- Draw an arrow in the direction of the solution. For \(x<3\), the arrow will point to the left.

ii) For **inclusive inequalities** like \(x≤3\) or \(x≥5\):

- Find the number.
- Fill in a solid circle, indicating that this value is part of the solution.
- Again, draw an arrow toward the solution side.

### Step 4: Two-Step Inequalities: Double the Fun, Double the Challenge

a) **Isolate the Variable**

i) For an inequality like \(2x−3>7\):

- Start by adding \(3\) to each side: \(2x>10\).
- Then, divide each side by \(2\): \(x>5\).

b) **Graph the Solution**

i) Much like one-step inequalities, locate the number on the line.

ii) Decide if you need an open or a solid circle based on the strictness or inclusiveness of the inequality.

iii) Draw the arrow pointing towards the solution’s direction.

### Step 5: Celebrate the Complexity: Compound Inequalities

Sometimes, you might find a beast like \(3<x≤8\). Here, \(x\) is trapped between two numbers.

- Graph the two numbers. One will have an open circle \((3)\) and the other, a filled one \((8)\).
- Connect them with a line or segment to show \(x\) can be any number in between.

### Step 6: Conclusion: Embrace the Dance of Inequalities

Understanding and graphing linear inequalities is like a dance of logic and intuition. With time and practice, you’ll find the rhythm and soon be swirling and twirling through even the most intricate inequalities with grace and precision!

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